Once the potential is known throughout Ω, then it is possible to calculate the current fluxes, I across each conducting boundary (in this case ∂Ω1 and ∂Ω2). At steady state the current flux across ∂Ω1 must be the additive inverse of the current flux across ∂Ω2,
I1 = Z
∂Ω1
σ∇Φ·dS=− Z
∂Ω2
σ∇Φ·dS . (2.36)
Once the current is calculated, it can be used in concert with the known potential difference across the region and Ohm’s law to determine an effective resistance.
Reff = V
I = |Φ1−Φ2| I1
(2.37)
In order to replicate this procedure for a spherical-particle-based model of the composite electrode, a number of assumptions and calculations beyond what are presented here are necessary. The full details of these calculations will be presented in Section2.4.
tions.
2. The important geometrical parameters associated with composite electrodes must be calculated from the resulting domain. The parameters of interest include the phase fractions, phase surface areas, TPB length, coordination numbers, percola- tion, and conductivity participation rates for the particles and connections within the domain.
3. The potential throughout the system must be calculated. In this case, the guiding assumption will be that the inter-particle distance and potential difference between connected particles will be small enough that the entire volume of each particle can be considered to be at one potential.
2.3.1 Domain Generation
2.3.1.1 Particle Placement
As discussed in Section2.1, most of the previous efforts to utilize particle-based models to describe composite anodes used the methodology introduced by Bennett [6]. Ben- nett’s approach places new particles just in contact with existing particles in the ensem- ble. Particle overlap and polydispersed particle size distributions can then be achieved by resizing the particles in place. This method is unsuitable for this work because we will consider the packing of particles whose sizes vary greatly, in some cases by orders of magnitude. Bennett’s method relies on an acceptance test for new particles. In the case when particles do not have a monodispersed size distribution, this method will result in a higher acceptance rate for smaller particles and a higher rejection rate for larger particles. Controlling the size distribution of the particle ensemble is extremely difficult using this method. To avoid this problem, a packing-sintering approach is used.
This method assigns a penalty function to each particle as it is serially added to the
ensemble. The penalty function achieves its minimum when the particle reaches the de- sired contact and overlap position with respect to its neighboring particles. Using this method, random close packing can always be established between particles, regardless of their relative size.
The first step is to place a particle at a random location, (xp, yp) in thex-yplane of the cermet box. The next step is to determine the largest z location at which the particle will contact an existing particle, or the z= 0 boundary of the box. To accomplish the first task we consider the locations in the x-y plane of all existing particles. Let the x-y location of the ith existing particle be represented by (xi, yi). The radii of the ith particle and of the new particle are ri and rp, respectively. The test particle contacts the ith particle if the sum of the radii is less than the distance between the particles,
(ri+rp)≤ q
(xi−xp)2+ (yi−yp)2 . (2.38)
If the set of particles satisfying this condition, U, is empty, then the test particle is placed at the location (xp, yp,0), as illustrated in Figure 2.9
As illustrated in Figure 2.10, if U is non-empty then the location, (xi, yi, zi), of the existing particle within U with the largest z value is used to determine the initial location of the test particle. The initialzlocation of the test particle, zp, is determined using equation 2.10
zp =zi+ q
(ri+rp)2−(xi−xp)2+ (yi−yp)2 . (2.39)
The location (xp, yp, zp) establishes a distance between the particles equal to the sum of their radii. A path is calculated from the initial contact point to the final position of the particle using a set of differential equations representing the force of gravity and inter-particle forces. The final resting point of the particle is found when either the
Figure 2.9: Initial placement of a non-intersecting particle
particle is a specified distance from each of its neighboring particles, or thezcoordinate of the particle is 0. A penalty function based on these conditions is used to calculate the acceleration of the particle, until the minimum is reached. The desired distance between two particles is determined by an overlap parameter,λ, and the radii of the two interacting particles. λparameterizes the length between the minimum and maximum possible distances allowed between two particles. The maximum distance between two contacting particles, dmax, is the sum of the radii of the two particles. The minimum distance is the point at which the radius of the circle of intersection between the particles is larger than the minimum of the two radii. This distance is:
dmin = q
|r12−r22|. (2.40)
Figure 2.10: Initial placement of an intersecting particle
The desired spacing between the two particles is then
d0 =dmin+λ·(dmax−dmin) . (2.41)
To place the particle at this spacing, the penalty function is used to define an acceleration vector for the particle. The path of the particle is integrated using the penalty function to define the acceleration vector of the particle. The function describing the inter- particle force between particles is:
F¯ = d−d0
2 . (2.42)
In addition, both a normal force ¯FN with respect to thez= 0 boundary, and a viscous damping force ¯FD are added to the differential equations in order to limit oscillatory behavior in the solution transients. The normal force between particles and the z = 0 boundary is:
F¯N=
prad ;pz > prad z ;pz ≤prad .
(2.43)
The damping force is the additive inverse of the velocity, ¯FD=−Vˆ.
The complete differential equation is:
dX¨¯ dt =
Np
X
i=1
¯ p−v¯i
k¯p−v¯ik ·d−d0
2 + ¯FN+ ¯FD . (2.44)
The final placement of the particle is demonstrated in Figure 2.11. Note that, as the boundaries of x-y plane are periodic, particles can translate through the x-z and y-z boundary planes.
Figure 2.11: Final particle placement
2.3.2 Particle Size Determination
Calculations are made using 4 different particle size distributions:
1. Mondispersed and simple polydispersed particle distributions.
2. Distributions calculated based on experimental data.
3. Integer-based discrete particle size distributions.
4. Truncated lognormal particle size distributions.
2.3.2.1 Monodispersed and Simple Polydispersed Particles
Calculating the properties of mondispersed and simple polydispersed distributions will allow comparison to previous computational studies on composite cermet calculations.
Because the filling problem is being solved in a different fashion than in previous studies it is important to consider the case of the simple monodispersed distribution, and the case in which the cermet is represented as a combination of two particle sizes, with each particle size assigned based on the type of particle represented. For structures with polydispersed particle distribution consisting of only two particle sizes assigned by phase, the ratio between particle sizes will be 1:0.781 YSZ:Ni. This ratio is based on the assumption that the precursor mixture for the composite anode is equal parts YSZ and nickel oxide (NiO) by volume, the percursor particles of YSZ and NiO are uniform and size, and there is a negligible amount of pore space in the precursor mixture. The manufacturing of the cermet involves the reduction of the NiO phase to Ni. The density of the precursor NiO is 6.67 g/cm3, and the density of pure Ni is 8.91 g/cm3.
2.3.2.2 Distributions Calculated from Experimental Data
Experimental measurement of precursor particles from Ni-YSZ composite anodes will be used to formulate a size distribution of particles for computation. The data will show non-analytical continuous size distributions. These distributions will typically contain particles with size differences of several orders of magnitude. In particular, studies will be based on particle arrays based on the particle size distributions described by Cho et al. [11].
0.1 1 10
Particle Size (%)
0 10 20 30 40
Cumulative Values (%)
NY-1 (Avg Diam: 0.70 µm) NY-2 (Avg Diam: 0.27 µm)
Figure 2.12: Particle size distributions published by Cho et al. [11]
Because these are empirical distributions with no known analytical form, there is no known analytical function which samples these distributions. These non-analytical dis- tributions are sampled using the Metropolis-Hastings (MH) [36,25] algorithm to sample the empirical distributions. The algorithm generates a Markov chain, the values of which have a distribution equivalent to the desired distribution. The process used in this study for drawing a sample from probability distribution, P, is described below.
1. A proposal distribution, Q, which can be directly sampled, is chosen. For this study, a lognormal distribution is used. The probability density of the lognormal distribution is
Q(x;µ, σ) = e−
(x−µ)2 2σ2
√
2πσ2 . (2.45)
The following function is used to generate a sample,x, with meanµ, and variance σ2, from the lognormal distribution
x= ln
1 +σ2 µ2
+xg·
s ln
1 +σ2
µ2
(2.46)
wherexg is a sample pulled from a normal distribution. The value of the param- eters µ and σ used were determined by a least-squares analysis comparing the lognormal distribution to the empirical distribution. This produced a lognormal distribution which matches more closely the empirical distributions, as compared to a lognormal distribution with the sameµand σ2 as the empirical distributions in this study.
2. The initial value of the Markov chain, x0, is set to a random value within the range of the empirical distribution. This becomes the current state of the Markov chain, xt.
3. Draw a value,x, randomly from the lognormal proposal distribution.
4. Calculate, the acceptance ratio, α,
α= P(x)·Q(xt)
P(xt)·Q(x). (2.47)
Then if α ≥ 1 then xt+1 = x; otherwise, xt+1 = x with probability α, and xt+1=xt with probability 1−α.
For this case a modified lognormal distribution is used as the test distribution which is applied for the Markov chain. Examples of MH-generated sample distributions com- pared to the Cho data are shown in Figures 2.13and 2.14.
The application of the MH algorithm used for this study has been simplified from the most general case, taking advantage of specific knowledge about the nature of this problem. While the lognormal distribution exists on the interval 0< x <∞, the desired probability distributions being reproduced exist on finite non-negative intervals of the form a ≤ x ≤ b. In order to disallow any values outside of the interval a ≤ x ≤ b, P(x;x < a, x > b) is set to the 1 × 10−14. This almost perfectly prevents any results which fall outside of the desired interval. Setting P(x;x < a, x > b) = 0 will result
Figure 2.13: Metropolis algorithm samples from particle size distributions published by Cho et al., 2008
α=∞, and could cause errors in any computer based code.
2.3.2.3 Lognormal Particle Size Distributions
The distributions used in Section 2.3.2.2do not have known analytical forms. For that reason it is not possible to parameterize these distributions and then examine the effect of changing parameters on important outputs. Lognormal distributions a set of well- understood, analytical probability distributions which can be easily parametrized. In addition, lognormal distributions are similar to the empirical distributions explored in Section 2.3.2.2. It was for this reason that lognormal distributions were chosen as the proposal distributions for the duplicating the empirical distributions via the Metropolis- Hastings methodology.
Figure 2.14: Metropolis algorithm samples from particle size distributions published by Cho et al., 2008
A lognormal distribution has the following probability density function, f,
f(x;µ, σ) = exp−
(ln(x)−µ)2
2σ2 . (2.48)
µis the mean of the logarithm of the variable,x, andσis the logarithm of the variance.
The mean, variance, and standard deviation of the lognormal distribution, respectively, are
E[X] =eµ+12σ2 ,
Var[X] = (eσ2 −1)e2µ+σ2 , Std.Dev[X] =p
Var[X] =eµ+12σ2p
eσ2 −1 .
The logarithm of samples from this distribution have a normal distribution. This allows a sample from a normal distribution,snorm, to be mapped to a sample from a lognormal distribution, sln, via
sln=snorm· s
ln
1 +σ2 µ2
+ ln(µ)− ln
1 +σµ22
2 . (2.49)
Methods for sampling a normal or Gaussian distribution can be found in a number of sources. For further information on the method used for this work, the reader is referred to pages 292–294 of Section 7.2 ofNumerical Recipes in C++ [39].
2.3.3 Calculation of Cermet Properties
After the computational domain has been generated and populated with particles, the next step is to calculate the geometrically-based parameters of the resulting virtual electrode. The geometrical parameters fall into two categories: Parameters which are primarily associated with the solid phases, and parameters which are primarily associ- ated with the void phase. These parameters are:
1. Solid Phase Parameters
(a) particle coordination numbers (b) TPB and connectivity
(c) relative phase volume
(d) surface areas of the two solid phases (e) inter-particle resistance
2. Void Phase Parameters
(a) porosity
(b) tortuosity of the void phase (c) mean pore diameter
The solid phase parameters and porosity are important to all the aspects of the modeling of composite electrodes that will be presented in this work. The void phase parameters, excepting porosity, are only relevant to the discussion in Chapter5.
2.3.3.1 Particle Coordination Numbers
All of the particles within the cermet are compared by location to find the intersecting particles within the cermet. Taking all pairs of particles pi and pj such that i > j where particle pi is located at the pointci = (xi, yi, zi) and has a radius,ri, a distance test determines whether or not the particles intersect. If the distance, d, between the respective particle centers is less than the sum of the particle radii, then the particles are assumed to intersect. The next step is to calculate the radius of the intersection, which using from the law of cosines and the Pythagorean theorem is
rintersection= v u u t
r2i − r2i +d2−r222
d2
. (2.50)
With the radius of intersection, both the area and circumference of the intersection can be calculated.
From this process three pieces of information are saved:
1. A list of all particles intersections along with the area of intersection,
2. A separate list of all heterogenous particle intersections along with the circumfer- ence of the intersection,
3. A list, for each particle, of the particles with which it intersects.
With the intersection information generated above it is possible to calculate the co- ordination numbers, Zx, of the cermet. The overall coordination number, Z, can be calculated by taking the total sum of intersecting particles for each particle in the en- semble and then dividing by the number of particles. The coordination numbers which are calculated are:
1. Z, Overall average coordination number.
2. Zi, Average coordination number for ionically conducting particles.
3. Ze, Average coordination number for electronically conducting particles.
4. Zi−i, Average coordination of ionically conducting particles with respect to ioni- cally conducting particles.
5. Ze−e, Average coordination of electronically conducting particles with respect to electronically conducting particles.
2.3.3.2 TBP Calculations
After the intersections of the cermet particles have been tabulated, the location and extent of the TPB, as well as the conductance between particles, are calculated. The nominal TPB length of the cermet is simply the sum of the circumferences calculated from the particle intersections, but a significant portion of that nominal length can be occluded by third particles. Calculating the actual TPB length is accomplished by discretizing the TPB between pairs of particles and determining what portion of that TPB is in fact interior to other particles in the cermet.
The first step is to determine the equation of the circle which describes the nominal TPB. The center point as well as two vectors in the plane of the circle are required.
The center point can be found via trigonometry. Letting ¯d be the vector ¯ci−c¯j, the center point of the intersection, ¯C, between two particles,pi and pj, is:
C¯ = ¯ci+ ¯d
ri2+kdk2−rj2
rikdk . (2.51)
Because any two vectors in the plane of the circle can be used for this calculation, there is no one method to determine the identity of the vectors. For this work the two vectors in the plane of the circle are generated as follows:
V¯1 = ( ¯dy,−d¯x,0), (2.52)
V¯2 = ¯d×V¯1 , (2.53)
where× is the vector cross product operator.
Theξ parametrized equation of the nominal TPB between intersecting particlespi and p2 is
P¯TPB(ξ) = ¯C+ ¯V1sinξ+ ¯V2cosξ, ξ∈[0,2π]. (2.54)
To assess the true length of the TPB, a set,N, of npoints (usually 100) are generated along the nominal TPB by allowing ξ = 2πn. Each of these points then represents a partial arc of the circle of length ∆s, where ∆s is the total perimeter of the circle divided by the number of points:
∆s= 2πr
n . (2.55)
Each point in N is evaluated as to whether that point is located within any particle intersecting eitherpiorpj. The recorded TPB length for the intersection betweenpi and
pj is then scaled by the fraction of points inN not located within any other particle. If multiple non-adjacent points are removed fromN via this process, the TPB generated by the intersection ofpi and pj is a set of arcs, defined by points within N.
The set of of remaining points within N define the TPB between particles pi and pj. The TPB can be visualized using the saved points. The length of the TPB segments generated by the intersection ofpiandpj is calculated as the fraction of points remaining inN multiplied by the circumference of the circle of intersection. The total TPB length for the cermet is then the sum of the scaled TPB lengths from each intersection between heterogenous particles.
The next step is to determine the connectivity of the TPB. The goal is to determine which segments of the TPB are in contact, as well as the frequency and extent of con- tacting networks of triple phase boundaries. The first step is determining the segments which will be used in the comparison process. The TPB segments determined using the procedure outlined above are not necessarily contiguous segments. The subtrac- tion of points can create segments which are composed of discontiguous arcs. Once the TPB segments are separated into segments which consist of contiguous sets of points, the distances between all points within the contiguous segments are calculated. If any calculated distance is less than the criterion distance,
∆s1+ ∆s2 , (2.56)
then the two segments are assumed to intersect and to be part of one contiguous curve.
Note that ∆s1 and ∆s2 are the curvilinear distance between points on the arc, which is longer than the actual Cartesian distance between the points. The criterion captures the distance between points in two intersecting arcs where the actual intersection occurs halfway between points which are used to define the arcs. Using this method there will
be a very small number of cases in each calculation where two curves which are not actually connected are projected to be connected.
2.3.3.3 Inter-Particle Resistance
The conductance of a homogenous volume with a constant conductivity, σ, a fixed cross-sectional area, A, and a length perpendicular toA,l, is:
G= σA
l . (2.57)
If the volume is assumed to be a sphere then A = 2πr and the conductivity can be restated as:
G= σπr2
l . (2.58)
Sunde used the results determined by Feng [17] to calculate the resistance between homogeneous particles in a disordered random sphere network. Given an intersection diameter between two homogeneous particles of 2δ the conductance between the two particles can be estimated as the conductance of a cylinder of diameter and length 2δ:
G= σπδ2
2δ = σπδ
2 . (2.59)
For the same calculation Schneider [42] uses an empirical relation developed by Argento and Bouvard [3]. This relation utilizes the particle radius, rp, and the intersection radius, rc. The assumption is that the base conductance between two particles, Gi, is determined by the conductance of a cylinder of radius rc and lengthrp:
Gi = σπrc2 rp
. (2.60)
The base conductance is adjusted depending on the ratio rc/rp:
G= 0.899 rrp
c
r 1−
rc
rp
2
Gi, for rrc
p <0.744
G=Gi, for rrc
p >0.744.
(2.61)
The initial approach used to determine the inter-particle conductance is a variation of the methods listed above. Given two homogenous intersecting particles with a radius of intersection, rc, and a distance between the particle centers, d, the conductance, G, is assumed to be equivalent to a cylinder with radius rc and lengthd.
G= σπr2c
d (2.62)
These results were then compared to a finite element solution, using Ansys, of the po- tential over a small set of particles in a cubic volume. The system, shown in Figure 2.3.3.3, contained 40 particles. The solution was obtained with a fixed potential dif-
Figure 2.15: FEM mesh and solution for particle array by Jay Sandal of Ansys [41]
ference across the cubic volume. In the particle model, particles are assumed to have